This isn't my own idea for an experiment, just a pointer towards an existing experiment that I couldn't find mentioned anywhere here.

It's an experiment being run by LaBerge's Lucidity Institute to study "Tibetan Dream Yoga," specifically how sleeping position affects dreams.

Dream Yoga says that you should "sleep on the right side as the lion doth" to help with dreams. These guys are investigating to see if there's any truth to it, figuring that maybe the nasal cycle (which nostril's easier to breathe through) affects dreams. It's already known to affect the nervous system, so it's not too far-fetched an idea.

Here's the link to the experiment description:
http://www.lucidity.com/DreamYoga.html

And the actual report to fill out and return to them:
http://lucidity.com/nosex2.pdf

The more data they collect, the better their results will be. Results need to be returned by December 31, 2007.